*Another way of visualizing if then statements is flowcharts.
Theorists may be interested in being able to make all the flowcharts
Non-theorists may be interested in the flowcharts they use continuing to work, but not too fussed about everything else.
Long:
It seems like there is a real phenomenon in computers and proofs (and some other brittle systems), where they are predicated on long sequences of precise relationships and so quickly break down as the relationships become slightly less true. But this situation seems rare in most domains.
Types of reasoning:
If x = 2, x^2 = 4. (Single/multi-case. Claims/if then statements.*)
For all x >= 0, x^2 is monotonic. (For every case. Universal quantification.)
Either can be wrong: the incorrect statement “If x =2, x^2 =3” and “For all real x, x^2 is monotonic.”
A claim about a single case probably isn’t a theory. Theories tend to be big.
We can think of there being “absolute theories” (correct in every case, or wrong), or “mostly correct theories” (correct more than some threshold, 50%, 75%, 90%, etc.).
Or we can think of “absolute”, “mostly correct”, “good enough”, etc. as properties. One error has been found in a previously spotless theory? It’s been moved from “absolute” to “mostly correct”. Theorists may want ‘the perfect theory’ and work on coming up with a new, better theory and ‘reject’ the old one. Other people may say ‘it’s good enough.’ (Though sometimes a flaw reveals a deep underlying issue that may be a big deal—consider the replication crisis.)
This post is about “absolute theories” or “theories which are absolute”. As paulfchristiano points out, a theory may be good enough even if it isn’t absolute. The post seems to be about: a) Critics need not replace the theory, and b) once we see that a theory is flawed, more caution may be required when using it, and c) pretending the theory is infallible and ignoring its flaws as they pile up will lead to problems. ‘If you see a crack in an aquarium, fix it, don’t let it grow into a hole that lets all the water out.’
The post might be about a specific context, where the author thinks people are ‘ignoring cracks in the aquarium and letting water out.’ Some posts do a bad job of being a continuation of one or more IRL conversations, but I found this to be a fairly good one.
Perhaps different types of theories should be handled differently, particularly based on how important the consequences are, and the difference between theories which are “absolute” and theories which are not, may matter a lot. If cars could run on any fuel except water, which would make them explode, but otherwise would be fine with anything...people wouldn’t just put anything in their fuel tank—they’d be careful to only put things that they knew didn’t have any water in them.
Or the difference mostly matters to theorists, and non-theorists are more interested in specifics (things that pertaining to the specific theories they care about/use), rather than the abstract (theories in general), and this post won’t be very useful to them.*
TLDR:
*Another way of visualizing if then statements is flowcharts.
Theorists may be interested in being able to make all the flowcharts
Non-theorists may be interested in the flowcharts they use continuing to work, but not too fussed about everything else.
Long:
Types of reasoning:
If x = 2, x^2 = 4. (Single/multi-case. Claims/if then statements.*)
For all x >= 0, x^2 is monotonic. (For every case. Universal quantification.)
Either can be wrong: the incorrect statement “If x =2, x^2 =3” and “For all real x, x^2 is monotonic.”
A claim about a single case probably isn’t a theory. Theories tend to be big.
We can think of there being “absolute theories” (correct in every case, or wrong), or “mostly correct theories” (correct more than some threshold, 50%, 75%, 90%, etc.).
Or we can think of “absolute”, “mostly correct”, “good enough”, etc. as properties. One error has been found in a previously spotless theory? It’s been moved from “absolute” to “mostly correct”. Theorists may want ‘the perfect theory’ and work on coming up with a new, better theory and ‘reject’ the old one. Other people may say ‘it’s good enough.’ (Though sometimes a flaw reveals a deep underlying issue that may be a big deal—consider the replication crisis.)
This post is about “absolute theories” or “theories which are absolute”. As paulfchristiano points out, a theory may be good enough even if it isn’t absolute. The post seems to be about: a) Critics need not replace the theory, and b) once we see that a theory is flawed, more caution may be required when using it, and c) pretending the theory is infallible and ignoring its flaws as they pile up will lead to problems. ‘If you see a crack in an aquarium, fix it, don’t let it grow into a hole that lets all the water out.’
The post might be about a specific context, where the author thinks people are ‘ignoring cracks in the aquarium and letting water out.’ Some posts do a bad job of being a continuation of one or more IRL conversations, but I found this to be a fairly good one.
Perhaps different types of theories should be handled differently, particularly based on how important the consequences are, and the difference between theories which are “absolute” and theories which are not, may matter a lot. If cars could run on any fuel except water, which would make them explode, but otherwise would be fine with anything...people wouldn’t just put anything in their fuel tank—they’d be careful to only put things that they knew didn’t have any water in them.
Or the difference mostly matters to theorists, and non-theorists are more interested in specifics (things that pertaining to the specific theories they care about/use), rather than the abstract (theories in general), and this post won’t be very useful to them.*